Switching employers on H1B

Gurus,

I need your help!

I started working at an OEM on (non IT) contract through Company X on OPT. Company X filed for my H1b last year and I got my H1B stamped in November. I am good to go for 3 years.
I have an offer from another smaller company Y as a direct employe and I really like the profile they are offering.

The new company Y is so small that I they haven't filed for an h1b before. According to their attorney I can start working for them as soon as the LCA is filed.

I want to know if I can do that. Earlier I thought that H1B transfer needs to be filed. I am worried a little because company Y does not have any experience in H1B.

I do not want any trouble later..

Could you please throw some light on transfer process and when I can actually start working with new employer?

No, You cannot .

Wait until official H1b transfer receipts are received from uscis. ( You will get a courtesy copy )

I would wait until the new H1 is approved. If there's time constraint in joining and you really want that job, spend extra for Premium Processing.